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HDP: 16 – 02
Physics of the Bacon Internal Resonator Banjo
David Politzer∗
(Dated: June 17, 2016)
The internal resonator banjo, patented and first sold by Fred
Bacon around 1906,
remains something of a cult favorite and is still produced by
some independent
luthiers. According to enthusiasts, the characteristic design
elements produce a
sound that is mellower, richer, and of greater complexity and
presence than without
them. Aspects of that sound are studied here, comparing
instruments that are
otherwise identical and identifying physics mechanisms that are
likely responsible.
∗[email protected];
http://www.its.caltech.edu/~politzer; Pasadena CA 91125
mailto:[email protected]://www.its.caltech.edu/~politzer
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Physics of the Bacon Internal Resonator Banjo
I. INTRODUCTION
In 1906 Fred Bacon, a virtuoso stage performer, patented[1] and
then formed a company
to produce and market what has come to be known as the internal
resonator banjo. In his
patent application he wrote:
“This invention has relation to certain improvements in the
construction of banjos or
other similar musical instruments whereby a more lasting tone is
produced and the quality
of same improved. The principal objection to the banjo resides
in the fact that the tones are
of short duration and that they therefore have a sharp staccato
quality which is objection-
able. The object of this invention is to overcome this objection
by providing the rim with a
peculiarly-constructed annular chamber within which the
partly-confined air can vibrate in
harmony with the strings and cooperate therewith to produce a
strong and resonant tone.”
The design has its enthusiasts to this day and is still produced
by independent luthiers.
When I was offered my first real job, I went shopping for my
first real banjo. (That was
1977. I had been playing a banjo I’d built from scratch when I
was 16.) A local music shop
carried some of the wonderful open-backs made by Kate Smith and
Mark Surgies. (They
were the A. E. Smith Banjo Company). However, my first real
paycheck was months away,
and I faced the expenses of moving across country and
resettling. Their top-of-the-line,
modeled on the Bacon Professional ff, seemed an extravagance. I
settled for their open-back
with a Bacon tone ring — a fine banjo by any measure. But I’ve
been fascinated by internal
resonators ever since.
To further my education in acoustics, a renowned expert strongly
recommended Rayleigh’s
book on sound.[2] At one point, I came across his illustration,
fig. 60, §310, reproduced here
as FIG. 2. Rayleigh imagined that a double Helmholtz resonator
might be somewhere of
interest. I imagined that it might be the key to the internal
resonator. I further realized
that the annular chamber might also support certain frequencies
of sound waves, not present
in its absence, just as Bacon said. It deserved closer
scrutiny.
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No. 823,985. PATENTED JUNE 19, 1906. F. J‘ BACON.
BANJO. APPLICATION FILED AUG.22. 1905.
FIG. 1. Page 1 of Bacon’s patent:
http://www.google.com/patents/US823985
http://www.google.com/patents/US823985
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FIG. 2. from Rayleigh’s Theory of Sound, fig. 60, §310
FIG. 3. The Bacon Professional ff internal resonator
FIG. 4. The double-Helmholtz interpretation of the internal
resonator
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5
II. THE BASIC PHYSICS IDEAS
A partial back and a cylindrical wall, extending from the back
to close to but not touching
the head, divide the cylindrical volume of the pot (the banjo
body) into an inner cylinder and
a surrounding annular region. The left-hand drawing in FIG. 4 is
a schematic representation
of a cross section of the pot. The two volumes are connected by
a constricted region formed
by the top of the wall and the tone ring. And the exit to the
outside air is a second
constriction formed by the partial bottom and the belly of the
player. The central drawing
of FIG. 4 is an idealization in terms of coupled Helmholtz
resonators. And the right-hand
figure is the mechanical analog: The black circles are the
masses of air in the constricted
regions, and the springs represent the compressible air in the
large volumes. The two dashed
circles are meant to represent ideal gears. They reflect the
fact that the two constricted
regions need not have the same area as they open into the
central cylinder. Hence, it has
the potential to act something like a hydraulic lift. Were this
picture applicable, there would
be two Helmholtz resonances, whose frequencies could be selected
over a very wide range by
adjusting the constriction dimensions. However, careful
measurements show that the actual
internal constriction is necessarily too large for this
interpretation, and there is really only
one Helmholtz resonance. In agreement with the basic features of
Helmholtz resonators, its
frequency is distinctly lower than that of a simple rim because
of the partial back.
Further experiments confirm that Bacon was right about the
higher frequency resonances
of the air in the pot. The combined system can be
well-represented as a coupling of two
well-understood systems: a smaller central cylinder and an
annulus of rectangular cross
section. While the problem of waves in an annulus is not exactly
soluble, it is certainly close
to a rectangular cross-section pipe with identified ends[3] — at
least for the dimensions
that appear in the internal resonator banjo. This coupled system
has a richer spectrum of
resonances, starting at a lower frequency than what is available
with the simple rim.
This report gives the details of these results and describes the
observations that support
them. Head motion is the primary producer of banjo sound. In
addition to the force of the
strings via the bridge, air motion inside the pot produces
pressure variations that push on
the head. So, banjo timbre is subtly effected by pot air
dynamics.
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III. ALTERNATE PHYSICS STRATEGIES
For many purposes, faithful mathematical modeling is of enormous
value. In such cases,
all available resources are brought to bear on the problem so
that the math description is as
detailed and accurate as possible. On the other hand, having a
simple way to think about
a system has its virtues. For some people, it’s just very
satisfying. But, even on the most
practical level, simple but valid pictures can help greatly with
creating novel designs and
finding solutions to particular practical problems. This project
is very much along the lines
of the simple pictures approach.
IV. OUTLINE
As already alluded to in the discussion pertaining to FIG. 4,
several lines of inquiry
did not pan out. The order of presentation herein reflects what
I think to be the simplest
explanation of the conclusions I finally drew. It is not the
actual order in which the study
unfolded.
Section V is an overview of Helmholtz resonators, and section VI
is a reminder and
summary of an accompanying, separate study of Bacon’s tone ring
on its own. Section
VII presents the evidence that there is only one Helmholtz
resonance (and not two) for the
internal resonator construction. Section VIII studies the air
vibrations where air sloshes
from place to place within the pot, identifying the particular
contribution of the internal
resonator annular region. Fully assembled banjos are compared in
section IX, including
plucks, spectra, spectrographs, and frailed sound samples. And
section X concludes with
the lessons learned.
V. HELMHOLTZ RESONATORS
Understood broadly, the concept of the Helmholtz resonance is of
enormous value in many
acoustical settings. There is, indeed, a simple formula that
applies to a very idealized situa-
tion. But interpreted qualitatively, that model suggests a way
to view and understand many
different, important systems. Helmholtz “invented” them to serve
as very sensitive, narrow-
band detectors of sound. However, they have been used to absorb
sound in architectural and
engineering settings and to produce sound in musical settings
since time immemorial. For
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7
most stringed instruments, they enhance the sound of the lowest
frequencies. The common
feature is that a relatively small volume of air is pushed back
and forth by the expansion
and contraction of a much larger volume.
N
shoulder of area Aneck of length Land area A and
volume V
enclosed air of volume V
FIG. 5. Helmholtz bottle resonators: real & ideal
In the idealized version, the volume, VN , of air in the neck of
the imagined bottle (see
FIG. 5) determines the mass of the oscillator. The larger
volume, V , produces the springi-
ness. The Hooke’s Law “spring constant” is actually proportional
to A2/V , where A is the
area of the interface. (If the bottle neck is cylindrical, then
VN = A × L, where L is the
length of the neck.) The resonant frequency, fH , is given
by
fH =vs2π
√A2
V×VNwhere vs is the speed of sound — which characterizes the
inherent springiness of air. To the
extent that this idealization is applicable, these are the only
parameters that enter into the
determination of fH . In particular, variations in shape and
positioning have no appreciable
effect.
The qualitative features of this formula account for the
behavior of any situation where a
relatively large, enclosed volume opens onto the open air
through some sort of constriction.
But even the real bottle shown in FIG. 5 presents problems with
taking the numbers too
seriously. Its volumes can be measured with water and a
measuring cup. The diameter of
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8
the cylindrical neck can be measured with a ruler. But how long
is the neck, and what is
the value of area A? For a long pipe, it is known and
qualitatively understood that the
pipe length effectively extends into the open air by 0.3 to 0.4
times the pipe diameter. That
“extra” length represents the mass of outside air that
participates significantly in the back-
and-forth motion. If the “neck” is, in fact, a stringed
instrument sound hole, even a rough
estimate of the effective VN presents a significant challenge.
And what about the bottom
of the neck? The real bottle in FIG. 5 has a smooth, curving
transition. That introduces a
small ambiguity in VN but a rather large ambiguity in A. The
effective A is clearly larger
than the cross section of the strictly cylindrical upper portion
of the neck — but not by too
much.
The pitch of the sound produced by blowing across the opening of
the particular bottle
in FIG. 5 was 122 Hz. Using water, a measuring cup, and a ruler,
adding the canonical
open-end correction, and making an educated guess for A, I got
127 Hz from the formula.
Of course, there exists a value for A for which the formula
gives exactly the right answer,
but, working a priori, one can only guess what that might
be.
The bottle as a musical instrument is typically tuned by fine
adjustment of V , e.g., by
partially filling the bottle with water.[4]
The qualitative dependence of fH on A is used in many musical
instruments. The ocarina
and its cousins are examples. (See FIG. 6.) Their frequency
spectra contain very little
besides the basic Helmholtz resonance. Enlarging the escape area
raises the pitch.
In the context of banjos, enlarging VN lowers the pitch. In the
design and set-up of
standard resonator banjos, attention is paid to this detail. In
the context of open-back
banjos, not only can VN be made bigger but A can be made
smaller. Both are accomplished
by a partial back. FIG. 7 shows a 100 year-old six-string banjo
with just such a back. Note
that the player’s belly is a crucial part of the open-back
banjo’s Helmholtz resonator.[5] V
is the volume inside the pot, VN is roughly the volume between
the partial back and the
player’s belly, and A is the region between the two V ’s.
Lowering the standard open-back
Helmholtz frequency was clearly the goal of the builder of this
100 year-old banjo to support
the sound of the additional low string.
For a banjo, sound production from the Helmholtz resonance is
not just the in-and-out
air. It is also heard in its effect on motion of the head (which
makes most of the sound)
as that motion is altered by the up-and-down pressure of the
Helmholtz resonance from
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FIG. 6. Ocarinas: Chinese xun, Bryan Mumford’s Puny Tune, and
Darryn Songbird’s Sweet
Potato (left) & bare hands (right)
underneath.
VI. THE BACON TONE RING
Bacon included a metal tone ring in his design. This item, shown
in cross section in
FIG.s 4 and 8, sits on the top edge of the wood rim, and the
drum head is stretched over
it. A detailed study of its vibrations and their effect on the
banjo’s sound is presented in
ref. [6]. In summary: the 1/4′′ solid diameter core hardens the
rim edge to reduce high
frequency absorption relative to a pure wood rim. The two thin
flanges stiffen the rim (at
the cost of very little extra weight) to reduce large rim
motions that otherwise absorb low
frequencies. And vibrations of the free horizontal flange absorb
some of the vibrational
energy that without the flange would have gone into sound.
In the present study, the focus is on air vibrational modes of
the pot. The tone ring
is considered, at least theoretically, as simply a fixed (and
ultimately irrelevant) part of
the defining geometry. Almost all of the comparisons are made
with a Bacon tone ring
installed on a single 11′′ Goodtime[7] rim. The pot geometry and
dimensions are altered
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FIG. 7. A 100 year-old, flush-fret six-string banjo
FIG. 8. The Steward-MacDonald reproduction Bacon tone ring,
discussed in some detail in ref. [6]
by attaching a variety of backs and internal resonators.
Whatever stiffening and vibrating
the tone ring does, it does so similarly for the different back
and internal geometries. Only
the comparisons in section IX involve an all-wood, standard
Goodtime rim, where it is
contrasted with a Bacon-like Goodtime, i.e., with tone ring and
full-size internal resonator
installed. This comparison involves all the effects at once, and
is, so to speak, the proof of
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the pudding.
VII. INTERNAL RESONATOR HELMHOLTZ RESONANCE
To disentangle the various mechanisms that may be at play with
an internal resonator,
first focus on the Helmholtz resonance(s). The situation is
further simplified by beginning
with partial backs with different size holes. The bottom of a
Bacon-tone-ring-equipped
Goodtime rim was cut flat and fitted with six threaded inserts.
The rings shown in FIG. 9
were cut from 3.0 mm, 7-ply birch and could be attached with a
narrow retaining ring of
the same plywood and six screws.
FIG. 9. Rings with various hole diameters that can be attached
to the back
With strings, neck, and tailpiece removed (but coordinator rod
in place), the sounds
of head taps with a piano hammer were recorded for various
bottom hole diameters. The
largest was the stock Goodtime, whose inner diameter is 9 3/4′′.
The smallest was 7 5/8′′,
which is the diameter of all the internal resonator inserts that
appear later.
I mounted a synthetic belly, made of closed cell foam, cork, and
Hawaiian shirt. (Those
materials mimic the absorption and reflection of a player’s
body). The opening to the outside
air is chosen to approximate typical playing and is far more
reproducible than holding the
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instrument up to one’s body. The genesis and details of this
back are discussed in ref. [5].
Since a Helmholtz resonance is characterized by motion of air in
and out of the sound hole,
I placed the microphone right at the opening. So the microphone
placement helps focus
on the specific modes of interest. Most of the sound actually
comes off the head and is
predominantly due to other modes.
FIG. 10 shows the spectra for long series of those head taps,
plotted for 100 to 500 Hz.
The vertical scale is decibels, a logarithmic measure of the
sound pressure. (Were the scale
linear in pressure, the individual resonances would look more
striking.)
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dB##$$>#
200$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$300$Hz$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$400$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$500$
Head#Taps#for#various#Back#Hole#Diameters#
9$3/4"$
9$1/4"$
8$1/2"$
7$5/8"$
FIG. 10. head taps with foam belly-back; curves labeled by back
hole diameter (no internal wall)
The strongest resonance for each back is the one with the second
lowest frequency. It
and the next lower one show a systematic decrease in frequency
with decreasing back hole
diameter. The higher resonances (at least six of them) show no
appreciable frequency
dependence on the back hole dimension.
This is precisely the qualitative behavior to be expected from
the Helmholtz formula.
Smaller hole diameter implies smaller A and larger VN in the
formula for fH . There are two
peaks for each back that reflect this behavior because the
internal pot Helmholtz resonance
couples strongly to the lowest drum mode of the head. Not only
do they both push the same
plug of air in and out, but they also push on each other over
the whole head surface. The
higher frequency modes are due to other physics. It is typical
that the lowest two modes
of the body of a stringed instrument are the coupled versions of
the Helmholtz and lowest
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13
sound board modes. And, on typical banjos, the fundamental
frequencies of all strings but
the high 5th are below 300 Hz. Also, on the banjo, the coupling
of the two low modes is
particularly strong because the head moves a lot compared to
wood sound boards.
FIG. 11. plywood heads, speakers, & mic
To separate Helmholtz from sound board physics on violins and
guitars, experimenters
have occasionally buried the instrument in sand – to immobilize
the sound board motion.
It’s easier on the banjo. I simply replaced the regular head
with 3/4′′ plywood. To drive
the Helmholtz resonance, I mounted a 3′′ speaker in the middle
of that plywood head. That
head is the one not attached to the rim in FIG. 11 and installed
on the rim in FIG. 12. (The
attached solid head and rim-mounted 1′′ speaker and microphone
in FIG. 11 are described
in section VIII.) The speaker is driven with a signal generator
and audio amplifier with a
slow sweep, logarithmic in frequency, over the desired
ranges.
So, the frequencies of the lowest two pot resonances are
substantially lowered by the
partial back in a way whose physics is qualitatively understood.
The next question is the
impact of the cylindrical wall of the internal resonator that
divides the interior into a smaller,
central cylinder and an outer annular volume. I fabricated a
variety of internal resonators,
shown in FIG. 13, all with the same cylinder diameter and back
hole size but with various
heights. The cylinders were cut from 3.0 mm, 5-ply maple drum
shell stock.
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14
FIG. 12. wood “head” with 3′′ speaker and cork & foam
belly-back
It turned out that what was really needed to reduce the physics
to something simple and
obvious were internal walls that came yet closer to the inner
surface of the plywood head.
Rather than fabricating a new set, I made a split ring that
could be inserted into the highest
original cylinder. It could be placed carefully at any
particular distance from the head when
assembled and tightened snugly with a shim in the gap in its
circumference. That is the
upper left construction in FIG. 13.
The resulting spectra for driving with the 3′′ head-mounted
speaker and listening with
a microphone at the rim-belly-back opening are plotted in FIG.
14. Now, for each pot
geometry, there is only one, low, broad peak between 200 and 300
Hz. With this set-up,
the higher resonances are all considerably weaker. All versions
are with the same rim with
its Bacon tone ring. The black curve, labeled “stock” refers to
the standard, open-back
Goodtime rim. The red curve, labeled “7 5/8′′” ring, is the
partial back with no cylindrical
wall. (That “7 5/8′′” is the same partial back hole size as all
of the internal resonators.)
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15
FIG. 13. internal resonators of various and variable heights
The blue and green curves refer to internal resonators that have
a 3/8′′ and 1/4′′ space,
respectively, between the top of the internal cylinder and the
inner surface of the head. The
pale blue “no gap” curve refers to an inner cylinder that
touches the head and seals off the
outer annulus from the inner cylinder.
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dB$$!!>$
300$Hz $ $ $ $$$ $ $ $ $1000$
Helmholtz)Resonance)Spectra)
stock$
7$5/8"$ring$
3/8"$gap$
1/4"$gap$
no$gap$
FIG. 14. plywood head with 3′′ speaker; foam belly-back
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The relations between the red, black, and pale blue curves are
standard Helmholtz res-
onator physics. The black (stock) and red (7 5/8′′ ring) have
that same V , but black has a
smaller A and a larger VN . The black and pale blue (no gap)
have the same A and VN , but
pale blue has a smaller V . Black and pale blue differ in all
three parameters; so the sign of
the difference depends on details of the actual values.
A very important lesson from these measurements, which was not
altogether obvious
beforehand, is that the Helmholtz resonances of the red, blue,
and green curves, i.e., all of
the pots with the same size partial back, are essentially
indistinguishable. That means that
the height of the internal cylindrical wall, going from zero all
the way up to the height in
the standard, finished banjo (i.e., reaching to 3/8′′ from the
inner surface of the head) does
not effect the frequency of the Helmholtz resonance. They are
all the same — as if there
were no inner wall at all. On one hand, the simple Helmholtz
resonance picture says that
the resonant frequency is independent of the shape of the
cavity. So, apparently, the wall is
simply an alteration in the shape. And these three
configurations have the same V , VN , and
A. On the other hand, one might ask whether there could be a
wall sufficiently high that it
divides the original cavity into two serial Helmholtz resonators
— just as Rayleigh suggested
could arise, at least for some design. Apparently, in practice,
the answer is no, not for the
internal resonator geometry. There are two obstacles. Friction
becomes an important force
with much smaller gaps. And the volume of the purported neck is
too small relative to the
interface area.
VIII. CAVITY MODES
The internal wall certainly does something, and that is revealed
by a study of the higher
frequency cavity modes. Again, the coupling to the head modes is
removed by using a solid
3/4′′ plywood head. That is the solid disk in FIG. 11. Since
these air modes are essentially
internal to the pot, the “sound hole” gap can be eliminated — to
allow for cleaner and
clearer resonances. The sound hole was only crucial to the
Helmholtz mode. So I chose
to seal the back with solid plywood. And that required putting a
driving speaker and a
recording microphone inside the pot. That assembly is shown in
FIG. 11.
Again, the rim is the Goodtime fit with a Bacon tone ring. Slow,
logarithmic frequency
scans, much as before but now with the small, internally mounted
speaker and microphone,
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17
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|̂#
dB#
Hz##$$>#
Cylinder#Internal#Resonator#Modes#
stock$
3/8"$gap$
FIG. 15. Comparison of the stock pot & the internal
resonator with a 3/8′′ head-to-resonator-wall
gap; lines at the bottom denote frequencies calculated solely
from the actual pot dimensions
yielded the spectra shown in FIG. 15. Note that the horizontal
frequency scale is linear.
“Stock” refers to the standard rim. “3/8′′ gap” is the standard
internal resonator, whose
cylindrical wall is 2 1/4′′ high, which brings it to 3/8′′ from
the inner surface of the head.
The are no Helmholtz resonances in this configuration because
there is no in-and-out air
motion. The lowest closed cavity resonances are the ones
shown.
At least below 1800 Hz, the internal resonator resonances appear
to have been shifted
lower, and there are more of them. The challenge is to
understand their origin.
The first step is to recognize that the stock pot resonances are
standard fare in under-
graduate physics and even are often used as a pedagogical
experiment in laboratory courses.
For an ideal cylinder the modes and frequencies can be
calculated. They are presented in
many places.[8] Pressure node lines and frequencies are shown in
FIG. 16 for the “squat”
cylinder. “Squat” means that the dimension perpendicular to the
circular cross section is
sufficiently small that there is no pressure variation in that
perpendicular direction until yet
higher frequencies where half a wave fits between the head and
the back.
The internal diameter of the actual pot was measured. The
resulting squat cylinder
frequency values are indicated by the black vertical lines at
the bottom of FIG. 15. (That
calculation won’t be perfect because it ignores the presence of
the speaker, microphone, tone
ring, and other hardware inside.) The additional grey lines,
first appearing around 2300 Hz,
are the calculated frequencies of the additional modes that
involve wave components in the
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18
FIG. 16. Pressure node lines of the lowest squat cylinder modes;
the number above is the frequency
in units of the lowest mode frequency
squat, perpendicular direction.
The inputs into the calculation are: the pot inner diameter and
height, the compressibility
of air (as encoded in the speed of sound), and Newton’s Laws.
The success is a triumph
of physics, but it is very old and well-known physics. The
“3/8′′ gap” purple line is the
spectrum we want to understand because it represents an
essential feature of the internal
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19
resonator banjo. Of course, it is not the final sound of the
banjo because those resonances
have to couple to the head; the net effects on the head and the
actual sound are subtle but
certainly within the range of things people can distinguish.
The key to conceptual progress came with abandoning the attempt
to relate the two
curves in FIG. 15 by building up the height of the wall. Rather,
an enlightening starting
point is a wall that leaves no gap between itself and the head.
Then the smaller inner
cylinder and the outer annular region are distinct. A small
acoustical coupling between the
two was introduced in the form of a 1/2′′ × 1/2′′ hole in the
internal wall. The result (and
more) is in FIG. 17.
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|̂#
dB#
Hz##$$>#
Cylinder#Internal#Resonator#Modes#
stock$
no$gap$
1/16"$gap$
1/8"$gap$
3/16"$gap$
3/8"$gap$
FIG. 17. Comparison of the stock pot & the internal
resonator with various gaps; lines at the
bottom denote frequencies calculated from the actual physical
dimensions
As in FIG. 14, the black curve comes from the stock pot scan,
and the purple curve is
for the 3/8′′ gap, i.e., the normal dimension internal resonator
with a 2 1/4′′ high wall. The
red curve is for the “no gap” wall that seals up against the
head.
I indicate on the bottom the calculated resonant frequencies for
the “no gap” system,
under the assumption that the coupling of its two parts is weak
enough to ignore. The long
dash orange/red lines are the standard cylinder mode
frequencies, higher than the black
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20
ones simply by the ratio of the stock diameter to the internal
resonator diameter, at least
for the squat, lower frequency range. The two cylinders have the
same height, and the higher
frequency contributions from waves in the perpendicular
direction are added in also.
The calculated mode frequencies of the annular volume are
indicated with short dash
pink/red and use the approximation described in section II. They
begin around 500 Hz,
which is substantially lower than the lowest mode of the stock
cylinder. Note that the
smallest dimension of the annulus is 1.01′′ in the radial radial
direction. That is only first
excited around 6700 Hz.
FIG. 17 also displays the measured spectra for intermediate
values of the rim-wall-head
gap, illustrating how the spectrum evolves continuously from no
gap to its final 3/8′′ value.
In contrast, the intermediate spectra resulting from
successively lower internal walls, i.e.,
going from the standard 2 1/4′′ height down to zero, which is
equivalent to the stock pot, are
more confusing than enlightening. (Remember that the back is
sealed with solid plywood;
there is no “partial back” in this part of the analysis.)
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dB$!!>$
$Hz$!!>$
Cylinder$&$Internal$Resonator$Modes$
stock&
1/2"&
1&1/4"&
2"&
2&1/4"&
FIG. 18. Spectra for different values of the internal cylinder
wall height
These are the spectra displayed in FIG. 18. The black and purple
curves do not quite match
those of FIG. 17. The origin of this discrepancy is the
following. Between taking the data
presented in each of the figures, the speaker and microphone
were removed and re-mounted,
without particular care to reproduce the original positions. The
resonant frequencies of the
pots should be unchanged, but their individual strengths can
differ significantly, particularly
at the higher frequencies. The point is that the speaker
subtends an angle of about 13o (in
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21
azimuth), and the microphone about 10o. Their separation was
about 26o. These numbers
can have a strong effect on how much a particular mode is
excited and how well it is detected.
IX. FULLY ASSEMBLED BANJO PLUCKED AND PLAYED
The following are a few comparisons of two fully assembled
banjos: a totally normal
Goodtime and a fully Bacon-like modified Goodtime, i.e., with
tone ring and internal 2 1/4′′
resonator. The strings, heads, and head tensions (as measured by
a DrumDial) were the
same.
Here is a sound recording and spectrograph of four typical
single string plucks, with the
other stings left free to vibrate. The microphone was at 20′′ in
front of the head. All plucks
were at the second fret. The first one is the 4th string of the
normal Goodtime; the second
is the 4th string of the Bacon-modified Goodtime; the third is
the 1st string of the normal
Goodtime; and the fourth is the 1st string of the Bacon-modified
Goodtime. (If your reading
is Web-enabled, the following links might be live; otherwise
they should be retrievable.) This
is the sound file:
http://www.its.caltech.edu/~politzer/bacon/4-plucks.mp3
and FIG. 19 is a spectrograph.
FIG. 19. Spectrograph of the four plucks described and linked
above
Before commenting on the differences, I offer here brief samples
of frailing on the two
http://www.its.caltech.edu/~politzer/bacon/4-plucks.mp3~
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22
banjos:
http://www.its.caltech.edu/~politzer/bacon/sample-A.mp3
http://www.its.caltech.edu/~politzer/bacon/sample-B.mp3
Which is which is revealed here: [10] .
A frequency spectrum analysis for each entire 35 second played
selection is displayed in
FIG. 20.
!85$
!75$
!65$
!55$
!45$
!35$
!25$100$ 1000$ 10000$
dB$$!!>$
1000$$Hz$
Internal(Resonator(vs.(Stock(Good3me(for(35(second(sound(samples(
stock$
internal$resonator$
FIG. 20. Spectra of 35 seconds of frailing on a Goodtime stock
vs internal-resonator-fitted banjo
The most obvious difference apparent in the FIG. 20 spectrum
analysis is in the power
below 200 Hz. In that range, neither is very loud in an absolute
sense. But the extra strength
provided by the Bacon design likely accounts for why some people
mention that the sound
of an internal resonator banjo is reminiscent of a 12′′ rim,
rather than an 11′′ (which is its
actual size).
The spectrograph helps to identify differences that can be heard
in the plucks and in the
actual played sound samples. Interestingly but not surprisingly,
there’s a lot going on. And,
gratifyingly, much can be traced to the particular physical
distinctions identified in this and
the accompanying tone ring study.
For the low open 4th string, the Bacon modifications make the
fundamental stronger but
the next six harmonics weaker. Then, above 1200 Hz, the
Bacon-style banjo has more power
and sustain. On the open 1st string, the Bacon is mostly
comparable or even a bit stronger
in power and sustain until around 1700 Hz, after which it’s the
other way around.
http://www.its.caltech.edu/~politzer/bacon/sample-A.mp3~http://www.its.caltech.edu/~politzer/bacon/sample-B.mp3~
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23
X. CONCLUSION
Apparently, the partial back and the annular region provide
lower body-air resonances
than any that are present on the stock, open-back banjo. And the
combined central cylinder
and annulus provide a richer spectrum of coupled resonances than
the simpler pot. Hearing
the effect of the lower resonances is straightforward.
The richer spectrum aspect is more subtle. Certainly, more
resonances, closer together,
give a more even response as a function of driving frequency.
But the representation of
a system by its spectrum loses all reference to time
development. And, for plucked (or
struck) strings, the time evolution is an essential
characteristic. Every aspect of the sound
is transient. However, the transients of coupled oscillating
systems are relatively neglected
in physics and engineering education and are not very widely
understood or appreciated.
Having resonances that are nearby in frequency but which arise
in different parts of a system
allow a variety of interesting phenomena.[9] The actual resonant
frequencies can depend on
the coupling strength. Energy can flow from one to the other and
back. This can extend
the lifetime or sustain. It can give rise to beats. So, the
internal resonator design not only
reduces some of the banjo’s most shrill bark, it increases at
least some of the sustain and
presents a richer, more complex sound.
NOTES
[1] Fred Bacon’s internal resonator patent:
http://www.google.com/patents/US823985
[2] John William Strutt, 3rd Baron Rayleigh, The Theory of
Sound, 1887 & 1896, still in print
and widely available in various formats. It’s not particularly
easy to read or use as a reference,
but it is dense with insights and one of the most inspiring
physics books I’ve ever encountered.
[3] “Identified ends” means that whatever goes out one end goes
in the other — and vice versa.
For example, the lowest frequency standing waves go around the
long way, with wavelength
equal to the circumference. The two node positions are not
determined.
[4] The Danish Bottle Boys are worth a listen. My on-line
favorite of theirs is
https://www.youtube.com/watch?v=NkbZlautuUc .
http://www.google.com/patents/US823985https://www.youtube.com/watch?v=NkbZlautuUc
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24
[5] D. Politzer, The Open Back of the Open-Back Banjo, HDP: 13 –
02,
www.its.caltech.edu/~politzer
[6] D. Politzer, A Bacon Tone Ring on an Open-Back Banjo, HDP:
16 – 01,
www.its.caltech.edu/~politzer
[7] I originally chose Deering Goodtime banjos for my acoustics
investigations because 1) they are
about as identical as wood objects can be, being a combination
of CNC fabrication and high
quality hand finishing; 2) they are quality instruments; and 3)
they are relatively inexpensive.
When I fist approached Greg Deering, requesting some special
items and perhaps a deal on
the price, he immediately offered to provide me with whatever I
needed. He has been of great
help ever since, including advice and fabrication related to
this project.
[8] e.g., M. J. Moloney, Plastic CD containers as cylindrical
acoustical resonators, Am. J. Phys.
77 (10) 882 (2009); DOI: 10.1119/1.3157150.
[9] D. Politzer,The plucked string: an example of non-normal
dynamics, HDP: 14 – 04, Am. J.
Phys. 83 395 (2015), doi:10.1119/1.4902310 or
www.its.caltech.edu/~politzer; Zany strings
and finicky banjo bridges, HDP: 14 – 05,
www.its.caltech.edu/~politzer
[10] B is the stock Goodtime, and A is a Goodtime fitted with a
bacon-style tone ring and internal
resonator.
~~~~
Physics of the Bacon Internal Resonator
BanjoAbstractIntroductionThe basic physics ideasalternate physics
strategiesOutlineHelmholtz ResonatorsThe Bacon Tone RingInternal
Resonator Helmholtz ResonanceCavity ModesFully assembled banjo
plucked and playedConclusionReferences